Converting a Definite Integral to a limit of a Riemann Sum and converting a limit of a Riemann Sum to a Definite Integral This template can be used as a starter file for presenting training materials in a group setting. Sections Right-click on a slide to add sections. Sections can help to organize your slides or facilitate collaboration between multiple authors. Notes Use the Notes section for delivery notes or to provide additional details for the audience. View these notes in Presentation View during your presentation. Keep in mind the font size (important for accessibility, visibility, videotaping, and online production) Coordinated colors Pay particular attention to the graphs, charts, and text boxes. Consider that attendees will print in black and white or grayscale. Run a test print to make sure your colors work when printed in pure black and white and grayscale. Graphics, tables, and graphs Keep it simple: If possible, use consistent, non-distracting styles and colors. Label all graphs and tables. 8-R

Riemann Sum approximation for the Area of Region If a region is bounded above by f(x) and below by the x-axis at all points of the interval [a,b], then the area of the region bounded by x = a and x = b using Right-hand rectangles is Height of Rectangle Width of Rectangle

Some Curve

Area Under a Curve Find the area of the region bounded by y = f(x), the x-axis, x = a, and x = b. Approximate the area by creating rectangles of equal width whose endpoints are on f(x). n = # of rectangles i = interval Each left endpoint is on f(x) Each right endpoint is on f(x) Each method is called a Riemann Sum.

The Limit Definition for finding the area under a curve using a right hand Riemann Sum Formula Height of the rectangle Width of the rectangle

Limit of a Riemann Sum The definite integral of a continuous function f over the interval [a, b] denoted by is the limit of Riemann sums as the widths of the subintervals approach 0. That is Where is the value in the ith subinterval, and Is the width of the ith subinterval. And n is the number of subintervals

1) Express the definite integral as the limit of a Riemann Sum

2) Express the definite integral as the limit of a Riemann Sum

3) Express the limit of a Riemann Sum as a definite integral

4) Express the limit of a Riemann Sum as a definite integral

5) Express the limit of a Riemann Sum as a definite integral

6) Express the limit of a Riemann Sum as a definite integral

Summation Formulas Proof of these formulas are found in Appendix A of our book

7) Find the area beneath (above the x-axis) in the interval [1,3] using the limit definition of a Riemann Sum

Summation Formula

Home Work Use a section header for each of the topics, so there is a clear transition to the audience. Worksheet 8-R